\(X\)-elliptic operators and \(X\)-control distances
The paper deals with linear second-order operator \(L\) in divergence form with nonnegative characteristic form. \(L\) is assumed to be \(X\)-elliptic. It means the existence of a family \(X=(X_1,\dots ,X_m)\) of locally Lipschitz continuous vector fields on \(\mathbb{R}^N\) such that, for a suitable constant \(c>0\), \[ \frac{1}{c}{\sum_{j=1}^m} \langle X_j(x),\xi_j\rangle^2\leq {\sum_{j=1}^m} a_{ij}(x)\xi_i\xi_j\leq c {\sum_{j=1}^m}\langle X_j(x),\xi_j\rangle^2 \] for every \(x,\xi\in\mathbb{R}^N\). A general condition on \(X\) assuring a scale invariant Harnack inequality for the positive weak solutions to \(Lu=0\) is established.
locally Lipschitz continuous vector fields
scientific article; zbMATH DE number 1898324
scientific article